{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:50:08Z","timestamp":1776840608403,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,10,4]],"date-time":"2002-10-04T00:00:00Z","timestamp":1033689600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we are concerned with the estimation of integrals on the unit circle of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Subscript 0 Superscript 2 pi Baseline f left-parenthesis e Superscript i theta Baseline right-parenthesis omega left-parenthesis theta right-parenthesis d theta\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>\n                                  \u03c0\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>e<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mi>\n                                  \u03b8\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>\n                              \u03c9\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _0^{2\\pi }f(e^{i\\theta })\\omega (\\theta )d\\theta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by means of the so-called Szeg\u00f6 quadrature formulas, i.e., formulas of the type\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma-summation Underscript j equals 1 Overscript n Endscripts lamda Subscript j Baseline f left-parenthesis x Subscript j Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sum _{j=1}^n\\lambda _jf(x_j)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"omega left-parenthesis theta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c9\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\omega (\\theta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    related to the Jacobi functions for the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-1,1],<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    nodes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace x Subscript j Baseline right-brace Subscript j equals 1 Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{x_j\\}_{j=1}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and weights\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace lamda Subscript j Baseline right-brace Subscript j equals 1 Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{\\lambda _j\\}_{j=1}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in Szeg\u00f6 quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01337-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"683-701","source":"Crossref","is-referenced-by-count":15,"title":["Szeg\u00f6 quadrature formulas for certain Jacobi-type weight functions"],"prefix":"10.1090","volume":"71","author":[{"given":"Leyla","family":"Daruis","sequence":"first","affiliation":[]},{"given":"Pablo","family":"Gonz\u00e1lez-Vera","sequence":"additional","affiliation":[]},{"given":"Olav","family":"Nj\u00e5stad","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,10,4]]},"reference":[{"key":"1","volume-title":"The classical moment problem and some related questions in analysis","author":"Akhiezer, N. 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Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"1-3","key":"10","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/0377-0427(95)00125-5","article-title":"Characterization of orthogonal polynomials with respect to a functional","volume":"65","author":"Peherstorfer, Franz","year":"1995","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"11","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01337-0\/S0025-5718-01-01337-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01337-0\/S0025-5718-01-01337-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:56:43Z","timestamp":1776725803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01337-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,4]]},"references-count":11,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01337-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01337-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2001,10,4]]}}}